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Theorem opprvalg 13247
Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprval.1  |-  B  =  ( Base `  R
)
opprval.2  |-  .x.  =  ( .r `  R )
opprval.3  |-  O  =  (oppr
`  R )
Assertion
Ref Expression
opprvalg  |-  ( R  e.  V  ->  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. ) )

Proof of Theorem opprvalg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opprval.3 . 2  |-  O  =  (oppr
`  R )
2 df-oppr 13246 . . 3  |- oppr  =  ( x  e.  _V  |->  ( x sSet  <. ( .r `  ndx ) , tpos  ( .r `  x
) >. ) )
3 id 19 . . . 4  |-  ( x  =  R  ->  x  =  R )
4 fveq2 5517 . . . . . . 7  |-  ( x  =  R  ->  ( .r `  x )  =  ( .r `  R
) )
5 opprval.2 . . . . . . 7  |-  .x.  =  ( .r `  R )
64, 5eqtr4di 2228 . . . . . 6  |-  ( x  =  R  ->  ( .r `  x )  = 
.x.  )
76tposeqd 6252 . . . . 5  |-  ( x  =  R  -> tpos  ( .r
`  x )  = tpos  .x.  )
87opeq2d 3787 . . . 4  |-  ( x  =  R  ->  <. ( .r `  ndx ) , tpos  ( .r `  x
) >.  =  <. ( .r `  ndx ) , tpos  .x.  >. )
93, 8oveq12d 5896 . . 3  |-  ( x  =  R  ->  (
x sSet  <. ( .r `  ndx ) , tpos  ( .r
`  x ) >.
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
10 elex 2750 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
11 mulrslid 12593 . . . . . 6  |-  ( .r  = Slot  ( .r `  ndx )  /\  ( .r `  ndx )  e.  NN )
1211simpri 113 . . . . 5  |-  ( .r
`  ndx )  e.  NN
1312a1i 9 . . . 4  |-  ( R  e.  V  ->  ( .r `  ndx )  e.  NN )
1411slotex 12492 . . . . . 6  |-  ( R  e.  V  ->  ( .r `  R )  e. 
_V )
155, 14eqeltrid 2264 . . . . 5  |-  ( R  e.  V  ->  .x.  e.  _V )
16 tposexg 6262 . . . . 5  |-  (  .x.  e.  _V  -> tpos  .x.  e.  _V )
1715, 16syl 14 . . . 4  |-  ( R  e.  V  -> tpos  .x.  e.  _V )
18 setsex 12497 . . . 4  |-  ( ( R  e.  _V  /\  ( .r `  ndx )  e.  NN  /\ tpos  .x.  e.  _V )  ->  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  e.  _V )
1910, 13, 17, 18syl3anc 1238 . . 3  |-  ( R  e.  V  ->  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )  e.  _V )
202, 9, 10, 19fvmptd3 5612 . 2  |-  ( R  e.  V  ->  (oppr `  R
)  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
211, 20eqtrid 2222 1  |-  ( R  e.  V  ->  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2739   <.cop 3597   ` cfv 5218  (class class class)co 5878  tpos ctpos 6248   NNcn 8922   ndxcnx 12462   sSet csts 12463  Slot cslot 12464   Basecbs 12465   .rcmulr 12540  opprcoppr 13245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-tpos 6249  df-inn 8923  df-2 8981  df-3 8982  df-ndx 12468  df-slot 12469  df-sets 12472  df-mulr 12553  df-oppr 13246
This theorem is referenced by:  opprmulfvalg  13248  opprex  13251  opprsllem  13252
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